Inverse Trigonometric Functions: Differentiation and Integration

The inverse trigonometric functions are also known as the "arc functions". There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as $\sin^{-1}$, $\text{asin}$, or, as is used on this page, $\arcsin$. Inverse trigonometric functions include $\arcsin$ and $\arccos$; $\arctan$ and $\text{arccot}$; and $\text{arcsec}$ and $\text{arccsc}$. They can be thought of as the inverses of the corresponding trigonometric functions.

Arcsine and Arccosine

The usual principal values of the $\arcsin(x)$ and $\arccos(x)$ functions graphed on the Cartesian plane.

Arctangent and Arccotangent

The usual principal values of the $\text{arctan}(x)$ and $\text{arccot}(x)$ functions graphed on the Cartesian plane.

Arcsecant and Arccosecant

Principal values of the $\text{arcsec}(x)$ and $\text{arccsc}(x)$ functions graphed on the Cartesian plane.

The differentiation of trigonometric functions is the mathematical process of finding the rate at which a trigonometric function changes with respect to a variable.

The integration of trigonometric functions involves finding the antiderivative.

The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions.

$\displaystyle{\int \text{arcsec}( a x) dx = x \ \text{arcsec}(a x) - \dfrac{1}{a} \text{arctanh}\left(\sqrt{1-\dfrac{1}{a^2 x^2}} + C\right)}$

$\displaystyle{\int \text{arccsc}( a x) dx = x \ \text{arccsc}(a x) + \dfrac{1}{a} \text{arctanh}\left(\sqrt{1-\dfrac{1}{a^2 x^2}} + C\right)}$

$C$ is used for the arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.

The derivatives of the inverse trigonometric functions are as follows:

$\dfrac{d}{dx} \text{arccot}(x) = \dfrac {-1} {1 + x^2}$

$\dfrac{d}{dx} \text{arcsec}(x) = \dfrac {1} {x^2\sqrt{1 - x^{-2}}}$

$\dfrac{d}{dx} \text{arccsc}(x) = \dfrac {-1} {x^2\sqrt{1 - x^{-2}}}$

Note that some of these functions are not valid for a range of $x$ which would end up making the function undefined.